Bring 2 cups water, the ginger, fennel seeds, and cardamom seeds to a boil in a small saucepan. Add tea bags; remove from heat. Let stand 3 minutes. Remove bags with a slotted spoon and discard.
Add apricots, figs, and sugar to tea mixture in pan. Return to a boil. Reduce heat; simmer, partially covered, until fruits are tender and liquid is reduced to a thick syrup, 20 to 25 minutes. Compote can be served warm or at room temperature. Serve yogurt on the side, with a little of the syrup drizzled on top.